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International Conference on Unconventional Computation

UC 2008: Unconventional Computing pp 111–128Cite as

On Faster Integer Calculations Using Non-arithmetic Primitives

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5204))

Abstract

The unit cost model is both convenient and largely realistic for describing integer decision algorithms over + ,×. Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware, may lead to a considerable drop in complexity. We show a variety of concrete problems to benefit from such non-arithmetic primitives by presenting and analyzing corresponding fast algorithms.

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References

  1. Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., Miltersen, P.B.: On the Complexity of Numerical Analysis. In: Proc. 21st Annual IEEE Conference on Computational Complexity (CCC 2006), pp. 331–339 (2006)

    Google Scholar 

  2. AMD64 Architecture Programmer’s Manual. vol. 1: Application Programming, Publication #24592 (Revision 3.13, July 2007)

    Google Scholar 

  3. Bach, E., Shallit, J.: Algorithmic Number Theory. Efficient Algorithms, vol. 1. MIT Press, Cambridge (1996)

    MATH  Google Scholar 

  4. Baran, I., Demaine, E.D., Pǎtraşcu, M.: Subquadratic Algorithms for 3SUM. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 409–421. Springer, Heidelberg (2005)

    Google Scholar 

  5. Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  6. Bertoni, A., Mauri, G., Sabadini, N.: Simulations Among Classes of Random Access Machines and Equivalence Among Numbers Succinctly Represented. Ann. Discrete Math. 25, 65–90 (1985)

    MathSciNet  Google Scholar 

  7. Bshouty, N.H., Mansour, Y., Schieber, B., Tiwari, P.: Fast Exponentiation using the Truncation Operation. Computational Complexity 2, 244–255 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  8. Borwein, J., Borwein, P.: PI and the AGM. Wiley, Chichester (1987)

    MATH  Google Scholar 

  9. Bshouty, N.: Euclidean GCD algorithm is not optimal (preprint, 1989)

    Google Scholar 

  10. Bshouty, N.: Private communication (1992)

    Google Scholar 

  11. Caldwell, C.K., Cheng, Y.: Determining Mill’s Constant and a Note on Honaker’s Problem. Journal of Integer Sequences, article 05.4.1 8 (2005)

    Google Scholar 

  12. Cheng, Q.: On the Ultimate Complexity of Factorials. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 157–166. Springer, Heidelberg (2003)

    Google Scholar 

  13. Coppersmith, D., Winograd, S.: Matrix Multiplication via Arithmetic Progressions. Journal of Symbolic Computation 9, 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fiduccia, C.M.: An Efficient Formula for Linear Recurrences. SIAM J. Comput. 14(1), 106–112 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gajentaan, A., Overmars, M.H.: On a Class of \(\mathcal O(n^2)\) Problems in Computational Geometry. Computational Geometry: Theory and Applications 5, 165–185 (1995)

    MATH  MathSciNet  Google Scholar 

  16. Granlund, T., Montgomery, P.L.: Division by Invariant Integers using Multiplication. In: ACM SIGPLAN Notices, pp. 61–72 (June 1994)

    Google Scholar 

  17. Han, Y.: Deterministic Sorting in \(\mathcal O\)(n · loglog n) time and linear space. Journal of Algorithms 50, 96–105 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Intel®64 and IA-32 Architectures Software Developer’s Manual, vol. 2A. Instruction Set Reference, A-M (order no.253666, May 2007)

    Google Scholar 

  19. Jacobson, N.: Structure of Rings. American Mathematical Society Colloquium Publications 37 (1964)

    Google Scholar 

  20. Just, B., auf der Heide, F.M., Wigderson, A.: On computations with integer division. RAIRO Informatique Theoretique 23(1), 101–111 (1989)

    MATH  Google Scholar 

  21. Kirkpatrick, D., Reisch, S.: Upper bounds for sorting integers on random access machines. Theoretical Computer Science 28(3), 263–276 (1983)

    Article  MathSciNet  Google Scholar 

  22. Koiran, P.: Valiant’s Model and the Cost of Computing Integers. Computational Complexity 13, 131–146 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lürwer-Brüggemeier, K., auf der Heide, F.M.: Capabilities and Complexity of Computations with Integer Division. In: Enjalbert, P., Wagner, K.W., Finkel, A. (eds.) STACS 1993. LNCS, vol. 665, pp. 463–472. Springer, Heidelberg (1993)

    Google Scholar 

  24. Montgomery, H.L., Vaughan, R.C.: The large sieve. Mathematika 20, 119–134 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mansour, Y., Schieber, B., Tiwari, P.: The Complexity of Approximating the Square Root. In: Proc. 30th IEEE Symposium on Foundations of Computer Science (FOCS 1989), pp. 325–330 (1989)

    Google Scholar 

  26. Pritchard, P.: A sublinear additive sieve for finding prime numbers. Communications of the ACM 24, 18–23 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pratt, V.R., Rabin, M.O., Stockmeyer, L.J.: A Characterization of the Power of Vector Machines. In: Proc. 6th Annual ACM Symposium on Theory of Computing (STOC 1974), pp. 122–134 (1974)

    Google Scholar 

  28. Randolph, J.F.: Basic Real and Abstract Analysis. Academic Press, London (1968)

    MATH  Google Scholar 

  29. Ribenboim, P.: The New Book of Prime Number Records, 3rd edn. Springer, Heidelberg (1996)

    MATH  Google Scholar 

  30. Ribenboim, P.: My Numbers, My Friends. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  31. Schönhage, A.: On the Power of Random Access Machines. In: Maurer, H.A. (ed.) ICALP 1979. LNCS, vol. 71, pp. 520–529. Springer, Heidelberg (1979)

    Google Scholar 

  32. Shamir, A.: Factoring Numbers in \(\mathcal O\)(log n) Arithmetic Steps. Information Processing Letters 8(1), 28–31 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  33. Simon, J.: Division is Good. In: Proc. 20th Annual Symposium on Foundations of Computer Science (IEEE FoCS 1979), pp. 411–420 (1979)

    Google Scholar 

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Authors and Affiliations

  1. Heinz Nixdorf Institute, University of Paderborn, 33095, Germany

    Katharina Lürwer-Brüggemeier & Martin Ziegler

Authors
  1. Katharina Lürwer-Brüggemeier
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  2. Martin Ziegler
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Editor information

Cristian S. Calude José Félix Costa Rudolf Freund Marion Oswald Grzegorz Rozenberg

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Lürwer-Brüggemeier, K., Ziegler, M. (2008). On Faster Integer Calculations Using Non-arithmetic Primitives. In: Calude, C.S., Costa, J.F., Freund, R., Oswald, M., Rozenberg, G. (eds) Unconventional Computing. UC 2008. Lecture Notes in Computer Science, vol 5204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85194-3_11

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  • DOI: https://doi.org/10.1007/978-3-540-85194-3_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85193-6

  • Online ISBN: 978-3-540-85194-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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